Thirdly, the approximate solution is proposed to find the best solution in the neighborhood of the initial solution in Section 3.2. Secondly, an initial solution of KP is constructed on the basis of the solution of upper bound of E- kKP in Section 3.1. Firstly, an upper bound is presented based on the exact k-item knapsack problem E- kKP in Section 2. In this paper, a fast polynomial approximate solution is proposed for 0-1 knapsack problems based on the solution of its upper bound. The solution can be proved to be the optimal solution. In this paper, it is easy to get an approximate solution of KP which objective value is 6395122580. It is easy to find that the core of KP is, so this problem is difficult to tackle by the exact algorithm in. He found that algorithms solving some kinds of core problem may be stuck by difficult cores. The items not included in the core are certain to be selected or not in the optimal solution, while the items in the core are uncertain to be selected or not in the optimal solution. Pisinger gave an exact algorithm for KP which is based on an expanding core. More domain knowledge is necessary for better algorithm. Meanwhile, these methods for the solution to KP are generally time-consuming. However, these algorithms cannot guarantee the accuracy of the solution of KP. Nowadays, it is tend to combine different heuristics together in solving combinatorial optimization problem, such as mixed-variable differentiate evolution, self-adaptive differential evolution algorithm, two-stage cooperative evolutionary algorithm, cooperative water wave optimization algorithm with reinforcement learning, and cooperative multi-stage hyper-heuristic algorithm. Heuristics rules are adopted to decrease the calculation in searching accurate approximation, such as harmony search algorithm, amoeboid organism algorithm, cuckoo search algorithm, binary monarch butterfly optimization, cognitive discrete gravitational search algorithm, bat algorithm, and wind driven optimization. While PTAS for KP typically require only O( n) storage, all FPTAS are based on dynamic programming and their memory requirement increases rapidly with the accuracy ɛ, which makes them impractical even for relatively big values of ɛ. Increasing the radius k of the neighborhood can improve the accuracy of the approximate algorithm. Approximation algorithm is often k-neighborhood local search method.

Greedy algorithm is O( n) time complexity, but it is not an ɛ-approximate algorithm. Greedy algorithm, an approximate algorithm for KP, is on the basis of certain rules, such as the item being selected with priority over larger efficiency, larger profit, or smaller weight. Let’s convert our understanding of 0/1 knapsack into python code.It is very important to find a fast polynomial approximation for KP in practice.

The profit after including object should be greater as compared to when the object is not included.

The total weight after including object should not exceed the weight limit.

There are two conditions that should be satisfied to include object : How do we decide whether we include object in our selection?

Or we don’t include object in our final selection.

Either we include object in our final selection.

Now for each cell, we have two options :

We do this because the 0th row means that we have no objects and the 0th column means that the maximum weight possible is 0. Let’s start by setting the 0th row and column to 0. Value of the cell with index represents the maximum profit possible when considering items from 0 to i and the total weight limit as j.Īfter filling the table our answer would be in the very last cell of the table.

0 1 knapsack